Punctuation and capitalization

In North American Academic English, the apostrophe is typically used to indicate two things: 1) a contraction, or 2) the possessive case. The apostrophe doesn't usually indicate the plural form of a noun, except when forming the plural of a lower case letter (e.g., p's and q's).

A contraction is an informal means of shortening two words into one word. For example, it is becomes it's, have not becomes haven't, and you are becomes you're. The apostrophe indicates that letters have been removed to form the contraction.

The possessive case is most easily determined if you can turn the possessive into an 'of the' statement. For example, the textbooks of the student becomes the student's textbooks, and the feathers of the peacock becomes the peacock's feathers. The apostrophe indicates the possessive case. However, if the noun is an object, building, or a piece of furniture, no apostrophe is needed: the table leg. See the list below for how to form the possessive case for different types of nouns, following the conventions of North American Academic English:

• add 's to the singular form of the word: the boy's hat
• if the singular noun ends in s, add 's after the noun (American Psychological Association, 2020, p. 162): James's car
• add 's to plural nouns that don't end in s: the children's toys
• add 's to plural nouns that end in s: the students's schedules
• add 's to the end of compound nouns: mother-in-law's hat (though hats belonging to more than one mother-in-law would be mothers-in-laws's hats)
• add 's to the final noun to indicate joint possession: Mary, Bob, Phil, and John's assignment

Per the dominant approach in North American Academic English, an apostrophe isn't used::

• when indicating a range of years: the 1990s is correct; the 1990's is incorrect.
• with a possessive pronoun, because the pronoun already indicates possession: his books is correct; his' books is incorrect.
• after the plural form of a noun: the books are over there is correct; the books' are over there is incorrect.

Finally, if you're writing in North American Academic English, here are a couple of other conventions about apostrophes to keep in mind:

• It's is a contraction of it is: It's sunny today. Its is the possessive case, indicating that the noun that follows belongs to 'it': the company announced its plan. 
• They're is a contraction of they are: They're leaving at 4:00. Their is the possessive pronoun: Where did the students leave their books? There indicates a place or position: I want to go there.

Reference

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000